1549380232-Automorphic_Forms_and_Applications__Sarnak_

318 FREYDOON SHAHID!, LANGLANDS-SHAHID! METHOD

an irreducible admissible x - generic representation i:J of M = M(F), these exist m
complex functions 1(s, i:J, ri, 'lj;p ), 1 ::; i::; m, such that:
(1) If F and i:J satisfy the conditions of Proposition 3.8, then
(3.12) 1(s, i:J, ri, 'lj;p) = c:(s, ri · <p, 'lj;p )L(l - s, i\ · <p)/ L(s, ri · <p).
m
(2) Equation (3.11) holds (in the form IT 'Y(is,i:J,i\,"°?jjp)).
i=l
(3) 1(s,i:J,ri,'l/JF) is multiplicative under induction (to be discussed below).
( 4) Whenever i:J becomes a local component of a globally generic cusp from,
then 'Y's become the local factors needed in their functional equations.
Moreover 1), 3) and 4) determine the 1-functions uniquely.
What is multiplicativity? We will discuss this only in examples since the
general formulation is complicated. It simply says that the 1-functions are mul-
tiplicative under parabolic induction and is a consequence of multiplicativity of
intertwining operators (2.10) under that (cf. [Sh3]). This is very deep from the
point of view of Rankin- Selberg method and usually quite hard to prove. Here are
some examples:

Example 1 (cf. [Sh7]). Suppose G = Sp(2n+2t) and M = GLn x Sp(2t), where
n and t are positive integers. Write i:J = i:J 1 @T. Suppose M' = GLn 1 x ... x GLnk x
GLti x ... x GLte x Sp(2a), where n 1 + ... + nk = n and t 1 + ... + te +a= t. By
case Cn of {Sh2}, r 1 is equal to the tensor product of the standard representation of
GLn(<C) and S02t+i(<C).

k e ,,

If i:J^1 = Q9 i:Jj 0 Q9 i:J b 0 T^1 , then multiplicativity simply means that, if

then

(3.13)

j=l b=l

i:J C Ind i:J^1 0 1 ,

M'N'TG

j=lb=l
k
f'(S, i:Jj I X i:Jb, _,, 'l/Jp) II f'(S, i:Jj I X T, I 'l/Jp ).
j=l
If Pn is the standard representation of GLn(C), then r2 = A^2 Pn·
i:Jj as before, multiplicativity for r 2 means

(3.14)

k

f'(S,i:J1,A

2

pn,'l/JF) =II 'Y(s,i:Jj, A^2 pnj>'l/JF)

j=l

II ')' ( s' (J~ x (Jj ' 'ljJ F).

1:5_i<j9

With i:J 1 and

No more ri beyond r 2 shows up and this is the case for all the classical groups.
Equality of the dimension on both sides of (3.14) simply means the following trivial
identity:

(3.15)